Skip to main content
SpringerLink
Log in

Lyapounov Functions of Closed Cone Fields: From Conley Theory to Time Functions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We propose a theory “à la Conley” for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independent such as the existence of Lyapounov functions) the case of continuous vector-fields on manifolds, of differential inclusions, of Lorentzian metrics, and of continuous cone fields. We generalize to this setting the equivalence between stable causality and the existence of temporal functions. We also generalize the equivalence between global hyperbolicity and the existence of a steep temporal function.

Résumé

On développe une théorie à la Conley pour les champs de cones, qui utilise une notion d’orbites relaxées basée sur les élargissements de cones dans l’esprit de la géométrie des espaces temps. On travaille dans le contexte des champs de cones fermés (ou, ce qui est équivalent, semi-continus), avec des singularités. Ce contexte contient (pour les questions indépendantes de la paramétrisation, comme l’existence de fonctions de Lyapounov) le cas des champs de vecteurs continus, celui des inclusions différentielles, des métriques Lorentziennes, et des champs de cones continus. On généralise à ce contexte l’équivalence entre la causalité stable et l’existence d’une fonction temporale. On généralise aussi l’équivalence entre l’hyperbolicité globale et l’existence d’une fonction temporale uniforme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernal A.N., Sánchez M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243(3), 461–470 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bernal A.N., Sánchez M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43–50 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bernard P., Zavidovique M.: Regularization of subsolutions in discrete weak KAM theory. Can. J. Math. 65, 740–756 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chernov V., Nemirovski S.: Cosmic Censorship of Smooth Structures. Commun. Math. Phys. 320, 469–473 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Chruściel P., Grant J.: On Lorentzian causality with continuous metrics. Class. Quantum Gravity 29(14), 145001,32 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Chruściel P., Grant J., Minguzzi E.: On differentiability of volume time functions. Ann. Henri Poincaré. 17(10), 2801–2824 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Clarke F.: Optimization and Nonsmooth Analysis. Pathum Wan, SIAM (1990)

    Book  MATH  Google Scholar 

  8. Clarke F., Ledyaev Y., Stern R.: Asymptotic stability and smooth Lyapunov functions. J. Differ. Equ. 149(1), 69–114 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Fathi A.: Partitions of unity for countable covers. Am. Math. Monthly 104, 720–723 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fathi A., Siconolfi A.: On smooth time functions. Math. Proc. Cambridge Philos. Soc. 152, 303–339 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Fathi A.: Time functions revisited. Int. J. Geom. Methods Mod. Phys. 12(08), 12 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Geroch R.: Domain of dependence. J. Math. Phys. 11, 437–449 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Hawking S.: The existence of cosmic time functions. Proc. R. Soc. Ser. A 308(1494), 433–435 (1969)

    Article  ADS  MATH  Google Scholar 

  14. Hawking S.W., Sachs R.K.: Causally continuous spacetimes. Commun. Math. Phys. 35, 287–296 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  15. Hurley M.: Chain recurrence and attraction in non-compact spaces. Ergod. Theory Dyn. Syst. 11(4), 709–729 (1991)

    Article  MATH  Google Scholar 

  16. Hurley M.: Lyapunov functions and attractors in arbitrary metric spaces. Proc. Am. Math. Soc. 126, 245–256 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Laudenbach F., Sikorav J.-C.: Hamiltonian disjunction and limits of Lagrangian submanifolds. Int .Math .Res. Not. 4, 161–168 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Minguzzi, E.: The causal ladder and the strength of K-causality. II. Class. Quantum Grav. 25, 015–010 (2008)

  19. Minguzzi E.: Characterization of some causality conditions through the continuity of the Lorentzian distance. J. Geom. Phys. 59, 827–833 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Minguzzi E.: On the existence of smooth Cauchy steep time functions. Class. Quantum Gravity 33, 115001 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Minguzzi, E.: Causality Theory for Closed Cone Structures With Applications. arXiv:1709.06494

  22. Minguzzi E., Sánchez M.: The causal hierarchy of spacetimes. In: Alekseevsky, D.V., Baum, H. (eds.) Recent Developments in Pseudo-Riemannian Geometry, pp. 299–358. European Mathematical Society, Zürich (2008)

  23. Monclair, D.: Attractors in Spacetimes and Time Functions. arXiv:1603.06994 (2016)

  24. Müller O., Sanchez M.: Lorentzian manifolds isometrically embeddable in L n. Trans. AMS 363(10), 5367–5379 (2011)

    Article  MATH  Google Scholar 

  25. Pageault P.: Conley barriers and their applications: chain-recurrence and Lyapunov functions. Topol. Appl. 156(15), 2426–2442 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sánchez M.: Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch’s splitting. A revision. Mat. Contemp. 29, 127–155 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Sánchez, M.: A Note on Stability and Cauchy Time Functions. arxiv:1304.5797 (2013)

  28. Sämann C.: Global hyperbolicity for spacetimes with continuous metrics. Ann. Henri Poincaré. 17, 1429–1455 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Seifert H.: The causal boundary of space-times. Gen. Relat. Gravit. 1, 247–259 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Siconolfi A., Terrone G.: A metric approach to the converse Lyapunov theorem for continuous multivalued dynamics. Nonlinearity 20(5), 1077–1093 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Siconolfi, A., Terrone, G.: A metric proof of the converse Lyapunov Theorem for semicontinuous multivalued dynamics. Discrete Contin. Dyn. Syst. Ser. A 32(12), 4409-4427

  32. Sullivan D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Author notes

  1. Stefan Suhr

    Present address: Fakultät für Mathematik, Ruhr-Universität Bohum, Universitätsstraße 150, 44780, Bochum, Germany

Authors and Affiliations

  1. École Normale Supérieure, DMA (UMR CNRS 8553), PSL Research University, Université Paris-Dauphine, 45, rue d’Ulm, 75230, Paris Cedex 05, France

    Patrick Bernard & Stefan Suhr

Authors
  1. Patrick Bernard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Suhr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrick Bernard.

Additional information

Communicated by P. Chrusciel

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement 307062. Stefan Suhr is supported by the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the DFG.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bernard, P., Suhr, S. Lyapounov Functions of Closed Cone Fields: From Conley Theory to Time Functions. Commun. Math. Phys. 359, 467–498 (2018). https://doi.org/10.1007/s00220-018-3127-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3127-7

Search

Navigation